Integral using Fourier transform of a characteristic function

Question

I want to calculate

$\int_{\mathbb{R}} \dfrac{\sin(at) \sin(bt)}{t^2}dt$,

with $a,b$ positive real numbers.

I'm not sure but I think I could use the Fourier Transoform of the characteristic function of the interval $(-c ,c ) $ where $c$ is positive (I think it could work well). Howevere I don't know how to use correctly this fact. Any hints?

Answer 1

Maybe this can work:

Let me call $f_a (x)$ the characteristic function of the interval $(-a ,a)$.

Now, using the relation between $\sin(x) $ and $e^{ix}$, I can find that \begin{equation} \hat{f_a} (t) = \dfrac{2 \sin (at)}{t} \end{equation}

Moreover $f_a \in L^1(\mathbb{R})$ and so $\hat{\hat{f_a}}(x) = (2 \pi) f_a(-x)$. Hence, \begin{equation} \int_{\mathbb{R}} \dfrac{\sin(at) \sin(bt)}{t^2} dt= \dfrac{1}{4} \int_{\mathbb{R}} \hat{f_a} (t) \hat{f_b} (t) dt = \dfrac{1}{4} \int_{\mathbb{R}} {f_a} (t) \hat{\hat{f_b}} (t) dt = \dfrac{\pi}{2} \int_{- \min(a,b)}^{\min(a,)} dt = \pi \min(a,b) \end{equation}

Where I used the fact that $f_b (-x) = f_b(x)$.